In this appendix, we show an alternative method for finding collective decay rates for the special case of a linear chain of N qubits. This method is less efficient than simply finding the poles of the scattering parameters, but it provides further insight into the physics of the system. Specifically, this method shows that each collective decay rate corresponds to a specific coherent excitation of the system. This means that one can always prepare initial states that excited only one of the N collective decay rates.
Let us start with the following Ansatz for an initially excited system following the approach by [26] |ψ(t)i= N X j=1 αj(t)e−iΩt|eji+|χ(t)i, (93)
where |χ(t)i includes the photon contribution with the initial condition |χ(0)i = 0. Then, the relation betweenαj(t)is found in [26] and can be given in terms of the variables defined in this
paper as ˙ αj(t) =− N X k=1 J0eiθ|j−k|αk(t−L|j−k|), (94)
whereα˙j(t) denotes the time derivative ofαj(t). [26] uses a more general form of this equation
to calculate observables (upon pulse incidence). Here, we will take another approach and show that this equation can yield the collective decay rates. We also note that here we are not interested in the time evolution of|χ(t)i.
Since we are interested in the collective decay rates, we take the Markovian limit such that αk(t−L|j−k|)'αk(t) sinceL∼O(Ω−1) andt∼O(J0−1). Consequently, (94) can be written as a matrix equation
˙
where x(t) = [α1(t), . . . , αN(t)] includes the qubit excitation coefficients and (J)jk =J0eiθ|j−k| is the collective coupling matrix. The solution to this equation is trivial (as long as J is non- singular, which it is for θ6=nπ; forθ=nπ, the existence of BIC changes the time evolution as discussed in Section 3.2) and can be given as
x(t) = N
X
l=1
βle−0.5Γltξl, (96)
where ξl and Γl/2 are eigenvectors and eigenvalues of the matrix J, and βl are some complex
constants that can be determined by the initial conditions8. This expression ties back to (12c),
whereΓl are indeed the collective decay rates.
This derivation reveals an important property of collective decay rates: they have one-to-one correspondence with states (|ξli) in the qubit subspace. It is important to note that these states
are not guaranteed to be orthogonal, although they are distinct and span the qubit subspace. While [26] discusses the eigenvalues and eigenvectors of the coupling matrix and associates, in passing, eigenvalues with collective decay rates when considering a pulse scattering problem, here we proved this relationship, which is only accurate for the Markovian limit.
The eigenstates of the matrix are the basis states of the N distinct decay modes. Conse- quently, each collective decay rate corresponds to a physical decay mode. This leads to the following phase space picture: the initial coherent excitation of the qubits can be written in terms of a linear combination of decay mode basis states. Consequently, a decay mode can only be accessed if the overlap is nonzero9. Hence, there is always a specific coherent excitation
(i.e. one that overlaps perfectly with the corresponding eigenvector) of qubits that can excite a single decay mode only. We discuss this property in the next section in the context of pulse shaping by engineering collective decay rates. Moreover, this property also explains how decay rates can signal the existence of BIC and why the dimensionality (N −1) of BIC is linked to the number (N −1) of zero collective decay rates. Since for θ =nπ, the J matrix has N −1
zero eigenvalues, the corresponding subspace has dimensionality N−1and can be constructed with orthogonal basis states. We also infer that subradiant states become BIC in a continuous manner asθ approachesnπ.
On another note, combining the one-to-one correspondence of decay modes and |ξli with
the symmetric and anti-symmetric collective decay rates conjecture, we realize that the states |ξli have either even or odd parity such that P|ξli = ± |ξli. The even (odd) parity states
correspond to symmetric (anti-symmetric) states. The implication is easy to prove via proof by contradiction. Assume |ξ1i has both symmetric and anti-symmetric parts. Then, it can be decomposed into both parts and hence can excite certain symmetric and anti-symmetric modes (that have non-zero overlap with the symmetric/anti-symmetric part of |ξ1i), which contradicts the conjecture. The fact that the decay mode states, |ξli, are either symmetric or
anti-symmetric, and not a mixture of two, is intriguing. For now, we do not have a conclusive proof for this, although we believe that the highly special shape of theJ matrix might be the first step towards understanding this phenomenon.
We emphasize that finding the collective decay rates via this method is inefficient, since it requires diagonalization of a N ×N matrix. Using the transfer matrix method is efficient, since it eliminates the internal degrees of freedom and deals with only2×2matrices, as shown in Appendix B. For example, using the transfer matrix method, the collective decay rates for N = 30can be found almost instantly, whereas it is nearly impossible to diagonalize the coupling
8
Here, we assumed, for simplicity, that J has a non-degenerate spectrum, which does not affect any of the arguments that follows.
9It is important to clarify that the eigenstates are not necessarily orthogonal, so the overlap is not taken as
orthogonal projections, but according to the angles between eigenstates. This leads to the fact that a system prepared in a certain eigenmode has non-zero probability to be observed at another mode. Nonetheless, this is
matrix. This phenomenon illustrates clearly why real-space approach outperforms the existing methods [26] by a large margin. We also emphasize that this approach works only in the Markovian limit. To find non-Markovian collective decay rates, one needs to use the machinery of the real-space approach.